For the natural numbers, we can define a set, an element of that set (the 0 or 1) and a function on that set (the successor function as in Peano's Axioms). In my previous posting here, "Natural numbers embedded in other sets," I posted a 112 line formal proof that natural number-like structures are embedded in every set on which there is defined an injective, non-surjective function. In a similar way, I have been able to show integer-like structures are embedded in other, much simpler structures.

For the integers, we can define 3 sets (z, zright, zleft), an element common to each (the 0), and a bijection (next) on one of those sets (z) as well as its inverse (next').

In the notation of DC Proof 2.0 (with epsilon mapped to "@" and informal comments before each axiom), the 14 Peano-like axioms for the integers are: